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What Does Unit Vector Mean?

A unit vector for the vector (4, 3)

Unit vectors are vectors with a length of $1$ . All vectors have a corresponding unit vector. You can find it by using this formula:

Formula

Unit Vector

{(x_{1}, y_{1})}_{unit vector} = \frac{1}{| (x_{1}, y_{1}) |} (x_{1}, y_{1})

You also have unit vectors that sit along the axes. All vectors in the plane are made up of a combination of these.

Theory

The Unit Vectors Along the Coordinate Axes

The unit vector along the $x$ -axis: ${\vec{e}}_{x} = (1, 0)$ .
The unit vector along the $y$ -axis: ${\vec{e}}_{y} = (0, 1)$ .

A vector can always be expressed using the unit vectors

{\vec{e}}_{x}

and

{\vec{e}}_{y}

like this:

(a, b) = a {\vec{e}}_{x} + b {\vec{e}}_{y}

Example 1

Find the unit vector of $(3, 4)$ .

\begin{array}{l} {(3, 4)}_{unit vector} & = \frac{1}{| (3, 4) |} (3, 4) \\ = \frac{1}{\sqrt{3^{2} + 4^{2}}} (3, 4) \\ = \frac{1}{\sqrt{25}} (3, 4) \\ = \frac{1}{5} (3, 4) \\ = (\frac{3}{5}, \frac{4}{5}) \end{array}

Example 2

Write the vector $(- 3, 5)$ by using the unit vectors that sit along the two axes.

You know that the unit vectors of the axes are ${\vec{e}}_{x}$ and ${\vec{e}}_{y}$ , and that those are ${\vec{e}}_{x} = (1, 0)$ and ${\vec{e}}_{y} = (0, 1)$ . That means you can decompose the vector and find the expression you’re looking for. It will look like this:

\begin{array}{l} (- 3, 5) & = (- 3, 0) + (0, 5) \\ = - 3 (1, 0) + 5 (0, 1) \\ = - 3 \cdot {\vec{e}}_{x} + 5 \cdot {\vec{e}}_{y} \end{array}

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